how to detect the points on which Newton Raphon method will give a oscillating sequence.

Question

Example consider the function $x^3 -x/2 +1/4$.. we have a oscillation if we start from 0 or 0.5 in Newton Raphson Sequence. Why this oscillation pattern observed and how to detect it?

Answer 1

Somewhat more generally, suppose $f(x) = x^3 + a x^2 + b x + c$. The corresponding Newton iteration is $x_{n+1} = g(x_n)$ where $$g(x) = x - \frac{f(x)}{f'(x)} = \frac {2 x^3 + a{x}^{2}-c}{3 x^2 +2\,ax+b}$$ A $2$-cycle for this iteration is $(x, g(x))$ where $g(g(x)) = x$ but $f(x) \ne 0$. It can be verified that such a $2$-cycle occurs when $$20\,{x}^{6}+40\,a{x}^{5}+ \left( 27\,{a}^{2}+19\,b \right) {x}^{4}+ \left( 6\,{a}^{3}+27\,ab-5\,c \right) {x}^{3}+ \left( 9\,{a}^{2}b-5\, ac+8\,{b}^{2} \right) {x}^{2}+ \left( -2\,{a}^{2}c+5\,a{b}^{2}+bc \right) x-abc+{b}^{3}+2\,{c}^{2} = 0$$ In particular this occurs for $x = 0$ and $x = g(0) = -c/b$ when $-abc + b^3 + 2 c^2 = 0$ but $c \ne 0$, which is the case for your example ($a=0, b=-1/2, c=1/4$).

Exercise: find $a,b,c$ such that $0, g(0), g(g(0))$ forms a $3$-cycle.